Summary: Let \(G\) be a simple graph of order \(n\) and let \(\gamma_{\mathrm{gdR}}(G)\) be the global double Roman domination number of \(G\). In this paper, we give some upper bounds on the global double Roman domination number of \(G\). In particular, we completely characterize the graph \(G\) with \(\gamma_{\mathrm{gdR}}(G)=2n-2\) and \(\gamma_{\mathrm{gdR}}(G)=2n-3\). Our results answer a question posed by \textit{Z. Shao} et al. [J. Discrete Math. Sci. Cryptography 22, No. 1, 31--44 (2019; Zbl 1495.05241)].